ranaazad wrote:
Bunuel wrote:
4. If \(xyz\neq{0}\) is \((x^{-4})*(\sqrt[3]{y})*(z^{-2})<0\)?
(1) \(\sqrt[5]{y}>\sqrt[4]{x^2}\)
(2) \(y^3>\frac{1}{z{^4}}\)
\(xyz\neq{0}\) means that neither of unknown is equal to zero. Next, \((x^{-4})*(\sqrt[3]{y})*(z^{-2})=\frac{\sqrt[3]{y}}{x^4*z^2}\), so the question becomes: is \(\frac{\sqrt[3]{y}}{x^4*z^2}<0\)? Since \(x^4\) and \(z^2\) are positive numbers then the question boils down whether \(\sqrt[3]{y}<0\), which is the same as whether \(y<0\) (recall that odd roots have the same sign as the base of the root, for example: \(\sqrt[3]{125} =5\) and \(\sqrt[3]{-64} =-4\)).
(1) \(\sqrt[5]{y}>\sqrt[4]{x^2}\) --> as even root from positive number (\(x^2\) in our case) is positive then \(\sqrt[5]{y}>\sqrt[4]{x^2}>0\), (or which is the same \(y>0\)). Therefore answer to the original question is NO. Sufficient.
(2) \(y^3>\frac{1}{z{^4}}\) --> the same here as \(\frac{1}{z{^4}}>0\) then \(y^3>\frac{1}{z{^4}}>0\), (or which is the same \(y>0\)). Therefore answer to the original question is NO. Sufficient.
Answer: D.
Hello sir,
It was not mentioned in the question stem whether x, y and z are integers. In this question, if x, y and z were non-integers, wouldn't the procedure be different? Would you please explain?
Thanks! :D
This question has a trick in that it will be true for both integers and non-integers alike:
1. x^2 > 0 no matter x=integer or not. Similarly, \(\sqrt[4]{x^2}\) will be positive for x=integer or non-integer. Try it out. Let x = 1 , \(\sqrt[4]{1^2} = 1 > 0\) and if x= 0.5, \(\sqrt[4]{0.5^2} = 0.707 >0.\)
Thus it does not matter whether you take x = integer or a fraction.
Coming back to the question,
Per statement 1, as mentioned above, \(\sqrt[4]{1^2} = 1 > 0\) ---> \(\sqrt[5]{y} > 0\) -----> y >0. Thus the statement \((x^{-4})*(\sqrt[3]{y})*(z^{-2})<0\) will be false. Thus this statement is sufficient.
Per statement 2, z^4 > 0 whether z =1 or z= 0.5 ----> 1/(z^4) >0 ----> y^3 >0---> y>0 (satisfies for both, y =1 or y = 0.25). Thus the given statement \((x^{-4})*(\sqrt[3]{y})*(z^{-2})<0\) will be false. Thus this statement is sufficient.
As both the given statements are sufficient, the correct answer is D.
whenever you are in doubt, always test the given statements/conditions with different values (integers, fractions , etc). This way you will get a better picture.
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